发表机构
School of Cyber Science and Technology, Shandong University（山东大学网络空间安全学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对分裂机制下MDS可转换码在r^F<k^F<r^I参数范围的带宽成本，建立新的秩不等式并推导出由三种情况组成的紧致下界，通过三种显式构造证明其最优性。
AI 中文摘要
擦除码广泛应用于分布式存储系统以提供容错能力。一个 $[n,k]$ 擦除码将 $k$ 个数据符号编码为 $n$ 个编码符号，并将其分布在 $n$ 个存储节点上。一旦码参数固定，可实现的容错能力也随之固定。然而，存储节点的故障率可能随时间变化，动态调整码参数以适应这些变化可以大幅降低存储开销。受此观察启发，Maturana 和 Rashmi 引入了可转换码，它允许将具有冗余度 $r^I=n^I-k^I$ 的 $[n^I,k^I]$ 初始码转换为具有冗余度 $r^F=n^F-k^F$ 的 $[n^F,k^F]$ 最终码，同时保持所需的码性质。初始码和最终码均为 MDS 码的可转换码称为 MDS 可转换码。这类码尤其受到关注，因为 MDS 码在给定的存储开销下提供最大的擦除容错能力。已有若干工作建立了分裂机制下 MDS 可转换码带宽成本的下界和构造。然而，在参数范围 $r^F<k^F<r^I$ 内的紧致界仍然未知。在本工作中，我们为具有线性转换过程的稳定 MDS 可转换码建立了一族新的秩不等式，并针对这一剩余参数范围推导出一个由三种情况组成的带宽成本改进下界。我们通过给出三种显式构造（每种情况对应一种构造）证明了该下界是紧致的。
英文摘要
Erasure codes are widely used in distributed storage systems to provide fault tolerance. An $[n,k]$ erasure code encodes $k$ data symbols into $n$ coded symbols and distributes them across $n$ storage nodes. Once the code parameters are fixed, the achievable fault tolerance is also fixed. However, the failure rates of storage nodes may vary over time, and dynamically adapting the code parameters to these variations can substantially reduce storage overhead. Motivated by this observation, Maturana and Rashmi introduced convertible codes, which allow an $[n^I,k^I]$ initial code with redundancy $r^I=n^I-k^I$ to be transformed into an $[n^F,k^F]$ final code with redundancy $r^F=n^F-k^F$, while preserving the required code properties. Convertible codes whose initial and final codes are both MDS codes are called MDS convertible codes. They are of particular interest because MDS codes provide the maximum erasure tolerance for a given amount of storage overhead. Several works have established lower bounds and constructions for the bandwidth cost of MDS convertible codes in the split regime. However, the tight bound in the parameter range $r^F<k^F<r^I$ remains unknown. In this work, we establish a family of new rank inequalities for stable MDS convertible codes with linear conversion procedures and derive an improved lower bound on the bandwidth cost consisting of three cases for this remaining range. We prove that the bound is tight by presenting three explicit constructions, one for each case.
